Let $s \in \Sigma^*$ be a formal language string. Consider the automorphism group of $s$, defined to be the set of all permutations of positions of $s$ that leave $s$ fixed. For instance $G(abab) = ...

In the wake of my curiosity on this kind of things, I was thinking if there is an example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$. Every example I ...

Some definitions...
Definition 1: A polynomial $f(X_1,\dots ,X_d)$ is $t$-linear if the variables $X_1,\dots ,X_t,\; t\leq d$ appear in all monomials of $f$ and degree of $X_i,\; i=1,2,\dots ,t$ on ...

All rings are assumed to be associative and unital.
If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for ...

A ring $R$ is called with bounded index (of nilpotency) $n$ if $n$ is the smallest natural number such that $a^n=0$ for all nilpotent $a \in R$.
Now let $R$ be a commutatitve ring with bounded index ...

The general question:
Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification
$$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$
between the space of symmetric order ...

While perusing through the article "Algorithms for solving linear ordinary differential equations" by Winfried Fakler (a pdf can be found through a google search), I noticed Faker mentioning on page 2 ...

Assuming ZFC. We can make $(\mathbb{R},+)$ into a nontrivial(scaler multiplication is not identicaly zero) $\mathbb{C}$-module.
Now my questions are?
0.Is it consistent with $ZF$ that $\mathbb{R}$ is ...

An associative $K$-algebra A is called reduced (or often basic) if $A/rad(A)$ has no nilpotent elements. It can be shown that this is equivalent to that $A/rad(A)$ is a isomorphic to a direct sum of ...

A Horn clause in the language of a universal algebra is a disjunction of equations and of at most one inequality
("equation" and "inequality" are the terms used by A.Horn in his paper "On sencences ...

I call adjunction algebra a universal algebra with one binary operation denoted as the punctuation sign (;) "semicolon" (but I will be using only one space after it, not on both sides - to avoid going ...

I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...

The membership relationship "$\epsilon$" is foundational for set theory, in the sense that the axioms of any set theory are formulated in the language of "$\epsilon$". Naturally, the question arises ...

Please correct me if I'm wrong, but it seems to me that two and three dimensional axiomatic quantum field theory were constructed as follow: the wightman axioms were formulated in euclidean space via ...

Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split?
If not, is there an example?
...

Can we use Positivstellensatz (P-satz) below for a non-polynomial term?
P-satz:
Let $R$ be real closed field. Let $f,g,h$ be finite families of polynomials in $R[X_{1} ,...,X_{n}]$. Denote by P the ...

For three ideals $I, J$ and $K$ of a subring $R$ in a number field $L$,
does this equality hold in general?
$(I+J) \cap K = (I \cap K) + (J \cap K)$
I have no counterexample yet but I couldn't prove ...